NNFL 3 - Guru Nanak Dev Engineering College

CHAPTER 3

SUPERVISED LEARNING NETWORK

“Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa

Copyright 2011 Wiley India Pvt. Ltd. All rights reserved.

DEFINITION OF SUPERVISED LEARNING NETWORKS Training and test data sets

Training set; input & target are specified



PERCEPTRON NETWORKS



Linear threshold unit (LTU)

x1

x2

xn

...

w1

w2

wn

w0

wi xi

1 if wi xi >0f(xi)= -1 otherwise

o

{

n

i=0

i=0

n



PERCEPTRON LEARNING

wi = wi + wi

wi = (t - o) xi

wheret = c(x) is the target value,o is the perceptron output, Is a small constant (e.g., 0.1) called learning rate. If the output is correct (t = o) the weights wi are not

changed

If the output is incorrect (t o) the weights wi are changed such that the output of the perceptron for the new weights is closer to t.

The algorithm converges to the correct classification• if the training data is linearly separable is sufficiently small



LEARNING ALGORITHM Epoch : Presentation of the entire training set to the

neural network.

In the case of the AND function, an epoch consists of four sets of inputs being presented to the network (i.e. [0,0], [0,1], [1,0], [1,1]).

Error: The error value is the amount by which the value output by the network differs from the target value. For example, if we required the network to output 0 and it outputs 1, then Error = -1.



Target Value, T : When we are training a network we not only present it with the input but also with a value that we require the network to produce. For example, if we present the network with [1,1] for the AND function, the training value will be 1.

Output , O : The output value from the neuron.

Ij : Inputs being presented to the neuron.

Wj : Weight from input neuron (Ij) to the output neuron.

LR : The learning rate. This dictates how quickly the network converges. It is set by a matter of experimentation. It is typically 0.1.



TRAINING ALGORITHM Adjust neural network weights to map inputs to outputs.

Use a set of sample patterns where the desired output (given the inputs presented) is known.

The purpose is to learn to • Recognize features which are common to good and bad

exemplars



MULTILAYER PERCEPTRON

Output Values

Input Signals (External Stimuli)

Output Layer

AdjustableWeights

Input Layer



LAYERS IN NEURAL NETWORK The input layer:

• Introduces input values into the network.• No activation function or other processing.

The hidden layer(s):• Performs classification of features.• Two hidden layers are sufficient to solve any problem.• Features imply more layers may be better.

The output layer:• Functionally is just like the hidden layers.• Outputs are passed on to the world outside the neural

network.



ADAPTIVE LINEAR NEURON (ADALINE)

In 1959, Bernard Widrow and Marcian Hoff of Stanford developed models they called ADALINE (Adaptive Linear Neuron) and MADALINE (Multilayer ADALINE). These models were named for their use of Multiple ADAptive LINear Elements. MADALINE was the first neural network to be applied to a real world problem. It is an adaptive filter which eliminates echoes on phone lines.



ADALINE MODEL



ADALINE LEARNING RULE

Adaline network uses Delta Learning Rule. This rule is also called as Widrow Learning Rule or Least Mean Square Rule. The delta rule for adjusting the weights is given as (i = 1 to n):



Initialize• Assign random weights to all links

Training• Feed-in known inputs in random

sequence• Simulate the network• Compute error between the input and the

output (Error Function)• Adjust weights (Learning Function)• Repeat until total error < ε

Thinking• Simulate the network• Network will respond to any input• Does not guarantee a correct solution

even for trained inputs

USING ADALINE NETWORKS

Initialize

Training

Thinking



MADALINE NETWORK

MADALINE is a Multilayer Adaptive Linear Element. MADALINE was the first neural network to be applied to a real world problem. It is used in several adaptive filtering process.



BACK PROPAGATION NETWORK



A training procedure which allows multilayer feed forward Neural Networks to be trained.

Can theoretically perform “any” input-output mapping.

Can learn to solve linearly inseparable problems.



MULTILAYER FEEDFORWARD NETWORK

Inputs

Hiddens

Outputs

I1

I2

I3

I0

h0

h1

h2

o0

o1

Inputs

Hiddens

Outputs



MULTILAYER FEEDFORWARD NETWORK: ACTIVATION AND TRAINING For feed forward networks:

• A continuous function can be • differentiated allowing • gradient-descent.• Back propagation is an example of a gradient-descent

technique.• Uses sigmoid (binary or bipolar) activation function.



In multilayer networks, the activation function isusually more complex than just a threshold function,like 1/[1+exp(-x)] or even 2/[1+exp(-x)] – 1 to allow for inhibition, etc.



Gradient-Descent(training_examples, )

Each training example is a pair of the form <(x1,…xn),t> where (x1,…,xn) is the vector of input values, and t is the target output value, is the learning rate (e.g. 0.1)

Initialize each wi to some small random value

Until the termination condition is met, Do• Initialize each wi to zero• For each <(x1,…xn),t> in training_examples Do

GRADIENT DESCENT



Input the instance (x1,…,xn) to the linear unit and compute the output o

For each linear unit weight wi Do

• wi= wi + (t-o) xi• For each linear unit weight wi Do• wi=wi+wi



Batch mode : gradient descent w=w - ED[w] over the entire data D

ED[w]=1/2d(td-od)2

Incremental mode: gradient descent w=w - Ed[w] over individual training examples d Ed[w]=1/2 (td-od)2

Incremental Gradient Descent can approximate Batch Gradient Descent arbitrarily closely if is small enough.

MODES OF GRADIENT DESCENT



SIGMOID ACTIVATION FUNCTION

x1

x2

xn

...

w1

w2

wn

w0

x0=1

net=i=0n wi xi

oo=(net)=1/(1+e-net)

(x) is the sigmoid function: 1/(1+e-x)d(x)/dx= (x) (1- (x))

Derive gradient decent rules to train:• one sigmoid function E/wi = -d(td-od) od (1-od) xi• Multilayer networks of sigmoid units backpropagation



Initialize each wi to some small random value.

Until the termination condition is met, Do

• For each training example <(x1,…xn),t> Do• Input the instance (x1,…,xn) to the network and

compute the network outputs ok • For each output unit k

– k=ok(1-ok)(tk-ok)• For each hidden unit h

– h=oh(1-oh) k wh,k k• For each network weight w,j Do• wi,j=wi,j+wi,j where

– wi,j= j xi,j

BACKPROPAGATION TRAINING ALGORITHM



Gradient descent over entire network weight vector

Easily generalized to arbitrary directed graphs

Will find a local, not necessarily global error minimum -in practice often works well (can be invoked multiple times with different initial weights)

Often include weight momentum term wi,j(t)= j xi,j + wi,j (t-1)

Minimizes error training examples

Will it generalize well to unseen instances (over-fitting)?

Training can be slow typical 1000-10000 iterations (use Levenberg-Marquardt instead of gradient descent)

BACKPROPAGATION



APPLICATIONS OF BACKPROPAGATION NETWORK Load forecasting problems in power systems.

Image processing.

Fault diagnosis and fault detection.

Gesture recognition, speech recognition.

Signature verification.

Bioinformatics.

Structural engineering design (civil).“Principles of Soft Computing, 2nd Edition”

by S.N. Sivanandam & SN DeepaCopyright 2011 Wiley India Pvt. Ltd. All rights reserved.

RADIAL BASIS FUCNTION NETWORK The radial basis function (RBF) is a classification and

functional approximation neural network developed by M.J.D. Powell.

The network uses the most common nonlinearities such as sigmoidal and Gaussian kernel functions.

The Gaussian functions are also used in regularization networks.

The Gaussian function is generally defined as



RADIAL BASIS FUCNTION NETWORK



SUMMARY

This chapter discussed on the several supervised learning networks like

Perceptron, Adaline, Madaline, Backpropagation Network, Radial Basis Function Network.

Apart from these mentioned above, there are several other supervised neural networks like tree neural networks, wavelet neural network, functional link neural network and so on.



Education

NNFL 3 - Guru Nanak Dev Engineering College